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arXiv · 2201.11484

Small Sets in Union-Closed Families

David Ellis·Maria-Romina Ivan·Imre Leader
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Abstract

Our aim in this note is to show that, for any $ε>0$, there exists a union-closed family $\mathcal F$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $ε$ of the sets in $\mathcal F$. More precisely, we give an example of a union-closed family with smallest set of size $k$ such that no element of this set belongs to more than a fraction $(1+o(1))\frac{\log_2 k}{2k}$ of the sets in $\mathcal F$. We also give explicit examples of union-closed families containing `small' sets for which we have been unable to verify the Union-Closed Conjecture.

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David Ellis, Maria-Romina Ivan, Imre Leader. 2023-01-23. Small Sets in Union-Closed Families. https://doi.org/10.37236/11004

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